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There are a great many locations on the Great Wall that can be visited. The most common start point if from Beijing, however, I would disagree with the answer by Mark Mayo in that they are not all over crowded with tourists. Most tours head to the section along Badaling, near Beijing, and that is the bit to avoid. Though again, if timed right and going off ...

Well there were actually several stages of the wall, and it's now in various states of disrepair / ruin depending on where you go. Presumably you want to go where there's actually wall, as opposed to rubble.
One of the most common routes is ex-Beijing, but since you want less crowded, we'll skip that.
For something quite cool and different, try Shanhaiguan ...

As with everything else in China - if you go to a place that is opened for tourists, be prepared to see a lot of them. You can usually avoid the hordes by going on a weekday (as opposed to weekend), avoiding national holidays, going early in the morning and picking a place that is less-known (pick #3, #4, ... place as opposed to #1, #2 place). That being ...

There are a great many locations on the Great Wall that can be visited. Narrowing the choice to those within day trip or single overnight stay from Beijing, we get the following list. I've added my comments as to their merits.
Miyun County
Simatai ( 司马台; Sīmǎtái) a popular but remote. Quite far from Beijing. Currently (2011) this section is closed to ...

Author Peter Hessler visited some of the most remote parts of the Great Wall and seemed to do an excellent job of avoiding other tourists. He took measures to avoid catching the attention of the local government, though. In planning your trip, you might do well to read his account of the journey: Country Driving: A Journey Through China from Farm to Factory

When you land at 3:00 AM, you should probably take taxi. It'll cost you around $55. (I've checked the rates over a site that i own, so i'm not sure if i should share the URL. Because i don't want to get marked as SPAM)
When you land at 11:00 AM, you should probably take the bus and following are the choices taken from ...

Beijing airport shuttle starts from 5.30 AM, and there will be no buses at that time. You will need to take a taxi.
I haven't been to the part of Great Wll you mentioned but I've been to Badaling and Mutiyanyu. Both are about half an hour from a bus, but again, you will have to take a taxi. Even with a bus, you will have to take 2 buses to go from Beijing ...